论文标题
$ kp^n+1 $数字的原始测试和安全素数的概括和Sophie Germain Primes
A primality test for $Kp^n+1$ numbers and a generalization of Safe Primes and Sophie Germain Primes
论文作者
论文摘要
在本文中，我们为$ kp^n+1 $的整数提供了Proth定理的概括。特别是，一个只需一个模块化凸起的原始测试，而没有计算任何GCD的模块化测试。我们还提供两项测试，以增加证明$ kp^n+1 $数字的机会（如果确实是素数）。作为这些测试的推论，我们向三个整数属$ n $提供了仅通过证明$ a^{n-1} \ equiv 1 \ pmod n $（Fermat的测试）才能证明其原始性的家族。我们还概括了安全的素数，并将这些概括的数字定义为与SafePrimes相似的$ -SAFRIME（因为这些数字的$ N-1 $与SafePrimes相同），我们解决了有关这些数字的分布的一些问题，并提供了有关其相关数字$ 100 $ $ $ $ $ $ $ $ $ $ $的$ 100 $ $ $ $ $ $ $ $的求婚的猜想。素数。
In this paper, we provide a generalization of Proth's theorem for integers of the form $Kp^n+1$. In particular, a primality test that requires only one modular exponentiation similar to that of Fermat's test without the computation of any GCD's. We also provide two tests to increase the chances of proving the primality of $Kp^n+1$ numbers (if they are primes indeed). As a corollaries of these tests we provide three families of integers $N$ whose primality can be certified only by proving that $a^{N-1} \equiv 1 \pmod N$ (Fermat's test). We also generalize Safe Primes and define those generalized numbers as $a$-SafePrimes for being similar to SafePrimes (since $N-1$ for these numbers has large prime factor the same as SafePrimes), we address some questions regarding the distribution of those numbers and provide a conjecture about the distribution of their related numbers $a$-SophieGermainPrimes which seems to be true even if we are dealing with $100$, $1000$, or $10000$ digits primes.
